Chpter 1 VECTOR LGEBR
INTRODUCTION: Electromgnetics (EM) m be regrded s the stud of the interctions between electric chrges t rest nd in motion. Electromgnetics is brnch of phsics or electricl engineering in which electric nd mgnetic phenomen re studied.
PPLICTION EM principles find ppliction in vrious disciplines such s; micowves,ntenns electric mchines stellite communictios Bioelectromgnetics plsms,nucler reserch,fiberoptics
Sclr nd Vectors sclr is quntit tht hs onl mgnitude. Quntities such s time,mss,distnce,temperture,entrop h,electric potentil.
VECTORS vector is quntit tht hs both mgnitude nd direction. Vector quntities include velocit,force,displcement nd electric field intensit.
UNIT VECTOR vector hs both mgnitude nd direction. The mgnitude of is sclr written s or unit vector long is defined s vector whose mgnitude is unit nd its direction is long, tht is;
UNIT VECTOR n
vector in Crte coordintes m be represented s; (,, ) + +
VECTOR DDITION & SUBTRCTION Two vectors nd B cn be dded together to give nother vector C C + B Let (,, ) nd B(B, B,B ) C ( + B ) + ( + B ) + ( + B )
Vector subtrction is similrl crried out s; D B + (-B) D ( -B ) + ( -B ) + ( -B )
Vector lgebr
Vector lgebr ddition ssocitive lw +(B+C) (+B)+C commuttive lw +B B+ multipliction b sclr B B distributive lw ( B+C) B + C
POSITION ND DISTNCE VECTOR point P in Crte coordintes m be represented b (,,). The position vector r p (rdius vector)of point P is s the directed distnce from the origin O to P; i.e., r p OP + +
P (3,4,5) 0 Z5 X3 Y4 POSITION VECTOR OP3 + 4 + 5
Distnce Vector P r PQ r p Q 0 r Q r PQ r Q -r p
Emples: If 10 4 + 6 nd B2 +. Find: () The components of long (b) The mgnitude of 3-B (c) unit vector long + 2B
Solution: () The component of long is -4 (b) 3-B 3(10,-4,6)-(2,1,0) (30,-12,18)-(2,1,0) (28,-13,18) Hence;
3 B 28 2 + ( ) 2 13 + ( 18) 2 35.74
( c ) Let C + 2B (10,-4,6) + (4,2,0) (14,-2,6) unit vector long C is c C C 14 2 ( 14, 2,6 ) + ( ) 2 2 + 6 2
Emples:2 Points P nd Q re locted t (0,2,4) nd (-3,1,5).Clculte () The position vector P (b) The distnce vector from P to Q (c) The distnce between P nd Q (d) vector prllel to PQ with mgnitud of 10
Solution 2 () r p 0 + 2 + 4 (b) r PQ r Q r P (-3,1,5)-(0,2,4)(-3,- 1,1) (c) Since r PQ is the distnce vector from P to Q,the distnce between P nd Q is the mgnitude of this vector;tht is, d r PQ 9 + 1+ 1 3.317
(d) Let the required vector be,then Where 10 is the mgnitude of.since is prllel to PQ, it must hve the sme unit vector s r PQ or r QP. Hence,
-9.045-3.015 +3.015
VECTOR MULTIPLICTION When two vector nd B re multiplied,the result is either sclr or vector depending on how the re multiplied. Thus there re two tpes of vector multipliction: 1. Sclr (or dot) product:.b 2. Vector(or Cross)product:XB
DOT PRODUCT The dot product of two vector nd B,written.B is defined geometricll s the product of the mgnitudes of nd B nd the ine of the ngle between them..b B B
The Vector Field Emple The Dot product.b B B B in the direction of You need to normlie before the dot product.
B Where is the smller ngle between nd B. The result of.b is clled either the sclr product. Let (,, ) nd B (B,B,B ).B B + B + B
Two vectors nd B re sid to be orthogonl (or perpendiculr) with ech other if.b0 Note tht dot product obes the following; (i) Commuttive Lw;.BB. (ii) Distributive lw :.(B+C).B +.C. 2
(iii) lso note tht... 0... 1
Cross Product The cross product of two vector nd B,written X B, is vector quntit whose mgnitude is the re of the prllelopiped formed b nd B B B
The Cross Product B N B B B Emple B B B 2 : 3 B 1 : 4 2 5 B 13 14 16
The cross product hs the bsic following properties: (i) It is not commuttive: B B (ii) It is not ssocitive; ( B C )( B ) C (iii) It is distributive (B+C)XB +XC
(iv) 0 lso note tht;,,
Cross Product ug clic permuttion. - - -
Emple:1 Determine the dot product nd cross product of the following vectors. 2 + 3 4 B -1 5 + 6
SOLUTION The dot product is.b (2)(-1) + (3) (-5) + (-4)(6) -41 The Cross Product B is B [ (3)(6) (-4)(-5)] + [(-4)(-1) (2)(6)] + [(2)(-5) (3)(-1)] -2-8 -7
lterntivel B - B B 2 +8 + 7
Sclr Triple Product Given three vector,b nd C.(BC)B.(C)C.(B) If (,, ), B(B,B,B ) nd C(C,C,C ).(BC) B B B Z C C C
Vector Triple product For vectors,b, nd C, we define the vector triple product s (BC)B(.C)-C(.B) It should be noted tht (.B)C(B.C) But (.B)CC(.B)
Coordinte Sstems nd Trnsformtion. Crte Coordintes (,,) Circulr Clindricl Coordintes ( ρ,, ) Sphericl Coordintes ( r,, )
Crte Coordinte Consists of three mutull orthogonl es,,, nd point in spce is denoted s P( 1, 1, 1 ). The loction of the point is defined b the intersection of three plnes.
Z P( 1, 1, 1 ) Z 1 1 1 + +
The Crte Coordinte Sstem
The differentil surfces in crte coordintes Differentil Surfces ds dd ds dd ds dd ds dd d d d ds dd ds dd
Crte Coordintes (,,) The Rnges of the coordinte vribles,, re;
Vector Components nd Unit Vectors
The Clindricl Coordintes Sstem. Form b three surfces One is plne for constnt, 1 The net surfce is clinder centered on the is of rdius ρ The third surfce is plne perpendiculr to the plne nd rotte bout the is b ngle Unit vector ρ,, point in the direction of increg coordinte vlue.
Circulr Clindricl Coordinte Sstem ρ dρ ρ d d 1 - Unit Vector vr with The coordinte Since direction chnges 2 Dot Product ρ d d ρ dρ d d
Clindricl coordinte sstem + 1 2 2 tn ρ ϕ ρ ρ ρ ρ
Z ρ S C Q D B d ρd dρ ρ
ρ d ρ ρ d
Differentil norml res in clindricl coordintes. ρd ρ d d ρ d d ρ ρ d
In Clindricl Coordintes,differentil elements cn be found s follows; (1) Differentil displcement is given b di dρ + ρd + ρ d (2) Differentil norml re is given b ds ρ d d ρ dρd ρddρ
(3) Differentil volume is given b dv ρ d ρ d d
Circulr Clindricl Coordinte Sstem ρ () ρ () Dot Product ρ 2 + 2 ρ 0 tn + + ρ ρ + + ρ ρ ρ ( + + ) ρ ρ + ρ ρ () ρ ( ) 1 ( + + ) + () () ( + + ) ρ 0
Emple: () Determine the volume enclosed b clinder of rdius ρ nd the length L s well s the surfce re of tht volume. (b) The surfce re?
Solution: To determine the volume enclosed we integrte; V v dv ρ ρ 0 πρ 2 2π L L 0 0 ρdrdd 14243 dv
2π L 2π ρ 2π ρ S 0 0 ρ d d 14243 4 Sides + 0ρ 0 rd dr 14243 bottom + 0ρ 0 rd top dr 14243 2πρL + 2πρ 2
Sphericl Coordintes point P cn be represented s ( r,, ) in this coordinte sstem vector in sphericl coordintes is written s or ( r,, ) + + r r Where, nd r re unit vectors long -direction. r,, nd
The Sphericl Coordinte Sstem ( ) r ( ) ( ()) r r ( ) r 2 + 2 + 2 r 0 2 + 2 + 2 0 180 tn
+ + + + r 1 2 2 2 1 2 2 2 tn Sphericl coordinte sstem r ϕ r r r
The Sphericl Coordinte Sstem ( ) r ( ) ( ()) r r ( ) rdr d ( ) r ( ) r 2 dr d d d r 2 ( ) dr d d
The rnges of vribles re; 0 r 0 π 0 2 π
r d dr rd r d d
Z r d r rd Differentil rc length for constnt
Z r d r r d rd Differentil rc length for constnt
Differentil norml res in sphericl coordintes; r d r r d rd dr rd () (b) (C) dr
(1) The differentil displcement is dl dr + rd + r r d (2) The Sphericl Coordinte Sstem ds r 2 r rdrd d drd d r
The differentil volume is dv r 2 drd d
Emple: Determine the volume enclosed b sphere of rdius r
Solution: v dv v r r 0 π 0 2π 0 r 2 drd d 4 3 π r 2
The reltionships between the vribles (,,) of the Crte Coordinte Sstem. P (,,)P ( ρ,, ) ρ ρ ρ
The mgnitude of is; 2 2 ρ + + 2 The reltionship between,, nd re obtined geometricll from, ρ, ρ + ρ
+ + Let ρ ρ convert to crte coordinte. + +. ( + + ) ρ ρ ρ ( ) ( ) + + ) ρ (
ρ ρ ρ π 2 + ρ
. ( + + ) ρ ρ ( ) ( ) + + ) ρ ρ ( + ρ Z Z
r 0 0 0 0 0
Performing dot-product ρ ρ
Clinder to Crte Crte to Clinder ρ ρ 2 + 2 ρ tn 1
EXMPLES:4 Given point P(-2,6,3) nd Vector + (+),epress P nd in clindricl coordintes. Evlute t P in Crte nd Clindricl sstems.
Solution: t Point P: -2, 6, 3 Hence, ρ 2 + 4+ 36 2 6.32 6 2 1 1 tn tn 108.43 0 Z3
Thus, P(-2,6,3)P(6.32,108.43 0,3) In the crte stem, t P is 6 + For vector,, + 0, Hence the clindricl sstem + 0 1 0 0 0 0 ρ
ρ + ( + ) + ( + ) But ρ, ρ
ρ ρ 008 6. 0.9487 40 38 40 6 ( ) [ ] ( ) [ ] ρ ρ ρ ρ ρ ρ ),, ( 2 + + + + + 40 6 40 2, 2 6 tn 40 ρ Hence P t
Crte Coordintes to Sphericl Coordintes Sstem Z ρ r P(,, ) P( r,, ) P( ρ,, ) r r ρ ρ ρ
r + + + +
In Mtri form,the ),, ( ),, ( r r r r r r 0 ),, ( ),, ( 0 r r tion Trnsform Inverse The
Dot Product of unit vectors in sphericl nd crte coordintes sstem. r 0
2 2 r + + 2 tn 1 2 + 2 tn 1
r r r
Emple: Evlute t P in the sphericl sstem s in Emple 4
In the sphericl sstem ) ( ) ( ) ( or r + + + + + + + 0 0 r
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